C₂²×#9497 #72319241

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

direct product

C₂²×#9497

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#386
commutator	C₂⁴	C₂⁴×C₄
Frattini, cosocle	C₂⁵	C₂⁵
prime radical	C₂²×#2403	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

36617 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 20 points:

(1,2)(3,4)(6,9)(10,12)(13,14)(15,16)(17,20)(18,19);(1,2)(3,4)(5,6,8,10)(7,9,11,12);(7,11)(9,12)(13,15,17,14)(16,19,18,20);(1,2)(6,10)(9,12)(13,16)(14,15)(17,18)(19,20);(3,4)

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#245233
maximal normal subgroup of	C₂²×#245233

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2)(3,4)(6,9)(10,12)(13,14)(15,16)(17,20)(18,19), (1,2)(3,4)(5,6,8,10)(7,9,11,12), (7,11)(9,12)(13,15,17,14)(16,19,18,20), (1,2)(6,10)(9,12)(13,16)(14,15)(17,18)(19,20), (3,4));

C22×#9497 #72319241